C₇×#5450 #13060757

order	1344
isoclinism	isoclinic class of #5450
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₇ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center, hypercenter	C₇	#5450
commutator, cosocle, hypocenter	C₂⁴	C₂²×C₃×C₇
Frattini	C₂²	C₂²×C₇×A₄
Fitting	C₇×H(3,𝔽₄)	C₃
socle	C₂²×C₇	C₂²×A₄
LCS	G ▷ C₂⁴
UCS	C₁ ◁ C₇

140 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 2 generators on 23 points:

(1,2,3,4,5,6,7)(8,10)(9,13)(11,18,19,17,12,23)(14,22,16,21,15,20);(8,16,12,9)(10,22,18,21)(11,14,19,15)(13,23,20,17)

maximal subgroup of	ΓL(1,𝔽₈)×#5450, C₇×#11710731
maximal normal subgroup of	ΓL(1,𝔽₈)×#5450

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10046);